1Concepts

1.1Introduction

A simple continued fraction is a number of the form a+1/(b+1/(c+...)), possibly infinite.
They have the convenient property that terminating the expression at some point results
in a rational that is "close" to the whole expression, in the sense that there is no
rational closer with a smaller denominator. For instance,
14/3 = 4 + 2/3 = 4 + 1/(3/2) = 4 + 1/(1 + 1/2). When all the numerators are
1 like this, we can simply write (4 1 2). This process will always terminate
for rational numbers since it is equivalent to the Euclidean algorithm.

Real solutions to quadratic equations with integer coefficients have repeating continued
fractions. For example, √2 = (1; 2 ...) which is
1 + 1/(2 + 1/(2 + ...)). Other algebraics do not have a repeating
simple continued fraction.

Simple continued fractions are unique, given a few caveats. As you can see
from the 14/3 example, 4 + 1/(1 + 1/2) = 4 + 1/(1 + 1/(1 + 1)), which is to say
that (a b ... c) is the same as (a b ... c-1 1).
Additionally, interspersed zeroes can be put in such that
(a b 0 n c ...) is the same as (a b+n c ...). Lastly, the representation
of negative numbers is not unique, either. If B is a [possibly infinite]
continued fraction (a b c ...) then there are two ways to write -B. The first
is (-a -b -c ...) and the second is (-(a+1) 1 |b|-1 |c| ...).
This library prefers the second form.

Many common functions or numbers have a continued fraction representation where the numerators
are not all 1s, such as the continued fraction for pi which can be written
0 + 4/(1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + ...)))). Such representations are not unique.

1.2Precision

When considered as a regular continued fraction, terminating the continued fraction at some point
yields a rational n/d, called a "convergent." Then the absolute value of the difference between this
rational and the value of the full continued fraction is less than one divided the product of the
denominators of this convergent and the next best convergent.
This factor is a worst-case scenario, most continued fractions have convergents with much better
precision. In fact, the golden ratio φ = (1+√5)/2 is the worst case, constantly
staying just inside the bounds. Because of this, the golden ratio is sometimes called
the "most irrational" number; that is, the number that is hardest to approximate by rationals.

1.3The Arithmetic of Continued Fractions

Continued fractions are amendable to term-at-a-time arithmetic. Suppose we have a function
f(x) = (ax+b)/(cx+d), where x is a continued fraction of the form
x = t + 1/x’, x’ being the rest of the continued fraction. Then when
f consumes the leading term t, x goes to t+1/x’, which after multiplication to get f back
in the same form, has updated the coefficients. If the continued fraciton x is in standard form,
then we have guarantees about every term except the first nonzero term, namely, that it will lie
somewhere between 1 and infinity (a terminated continued fraction). Then if the limits of f
as x goes to 1 and as x goes to infinity agree it does not matter what the value of x’ is as
the leading term of f is already determined. We can take a quotient term out of the function, and
f then goes to t + 1/f’.

For further details, reference the MIT HAKMEM memo 239 (Gosper).

1.4Purpose

Because of the term-at-a-time aspect of continued fraction arithmetic, calculating terms and
quotients proceeds without unnecessary calculation. Because of the precision guarantees,
calculation can be terminated when the desired precision is exceeded. Combining these facts
give us a picture of exact arithmetic of rationals and irrationals which only calculate
as far as needed and yield the rational with the smallest denominator possible for the
given precision.